constrained optimization methods in health services ... · *corresponding authors – kalyan s....

ISPOR Optimization Methods Emerging Good Practices Task Force Report

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Constrained Optimization Methods in Health Services Research – An Introduction: Report

1 of the ISPOR Optimization Methods Emerging Good Practices Task Force

William Crown, PhD1,*, Nasuh Buyukkaramikli, PhD2, Praveen Thokala, PhD3, Alec

Morton, PhD4, Mustafa Y. Sir, PhD5, Deborah A. Marshall, PhD6, Jon Tosh, PhD7, William

V. Padula, PhD, MS8, Maarten J. Ijzerman, PhD9, Peter K. Wong, PhD, MS, MBA, RPh10,

Kalyan S. Pasupathy, PhD11,*

1 Optum Labs, Boston, MA, USA; 2 Scientific Researcher, institute of Medical Technology assessment, 1

Erasmus University Rotterdam, Rotterdam, The Netherlands; 3 Research Fellow, University of Sheffield, 2

Sheffield, UK; 4 Professor of Management Science, Department of Management Science, Strathclyde 3

Business School, University of Strathclyde, Glasgow, Scotland, UK; 5 Assistant Professor, Health Care 4

Policy & Research, Faculty - Information and Decision Engineering, Mayo Clinic Robert D. and Patricia 5

E. Kern Center for the Science of Health Care Delivery, Rochester, MN, USA; 6 Canada Research Chair, 6

Health Services & Systems Research; Arthur J.E. Child Chair in Rheumatology Research; Director, HTA, 7

Alberta Bone & Joint Health Institute; Associate Professor, Department Community Health Sciences, 8

Faculty of Sciences, Faculty of Medicine, University of Calgary, Calgary, Alberta, Canada; 7 Senior 9

Health Economist, DRG Abacus, Manchester, UK; 8 Assistant Professor, Department of Health Policy & 10

Management, Johns Hopkins Bloomberg School of Public Health, Baltimore, MD, USA; 9 Professor of 11

Clinical Epidemiology & Health Technology Assessment (HTA); Head, Department of Health Technology 12

& Services Research, University of Twente, Enschede, The Netherlands; 10 Vice President and Chief 13

Performance Improvement Officer, Illinois Divisions and HSHS Medical Group, Hospital Sisters Health 14

System (HSHS), Belleville, IL. USA; 11 Associate Professor - Healthcare Policy & Research, Scientific 15

Director – Clinical Engineering Learning Labs, Lead – Information and Decision Engineering, Mayo 16

Clinic Robert D. and Patricia E. Kern Center for the Science of Health Care Delivery, Rochester, MN, 17

USA. 18

*Corresponding authors – Kalyan S. Pasupathy ([email protected]) and William Crown

([email protected])

Abstract 19

Providing health services with the greatest possible value to patients and society given the constraints 20

imposed by patient characteristics, health care system characteristics, budgets, etc. relies heavily on the 21

design of structures and processes. Such problems are complex and require a rigorous and systematic 22

approach to identify the best solution. Constrained optimization is a set of methods designed to identify 23

efficiently and systematically, the best solution (the optimal solution) to a problem characterized by a 24

number of potential solutions in the presence of identified constraints. This report identifies: 1) key 25

concepts and the main steps in building an optimization model; 2) the types of problems where optimal 26

solutions can be determined in real world health applications and 3) the appropriate optimization 27

methods for these problems. We first present a simple graphical model based upon the treatment of 28

“regular” and “severe” patients, which maximizes the overall health benefit subject to time and budget 29

mailto:[email protected]

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constraints. We then relate it back to how optimization is relevant in health services research for 30

addressing present day challenges. We also explain how these mathematical optimization methods relate 31

to simulation methods, to standard health economic analysis techniques, and to the emergent fields of 32

analytics and machine learning. 33

Keywords: Decision making, care delivery, policy, modeling 34


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1. Introduction 35

In common vernacular, the term “optimal” is often used loosely in health care applications to refer to any 36

demonstrated superiority among a set of alternatives in specific settings. Seldom is this term based on 37

evidence that demonstrates such solutions are, indeed, optimal – in a mathematical sense. By “optimal” 38

solution we mean the best possible solution for a given problem given the complexity of the system inputs, 39

outputs/outcomes, and constraints (budget limits, staffing capacity, etc.). Failing to identify an “optimal” 40

solution represents a missed opportunity to improve clinical outcomes for patients and economic 41

efficiency in the delivery of care. 42

43

Identifying optimal health system and patient care interventions is within the purview of mathematical 44

optimization models. There is a growing recognition of the applicability of constrained optimization 45

methods from operations research to health care problems. In a review of the literature [1], note more 46

than 200 constrained optimization and simulation studies in health care. For example, constrained 47

optimization methods have been applied in problems of capacity management and location selection for 48

both healthcare services and medical supplies [2-5]. 49

Constrained optimization is an interdisciplinary subject, cutting across the boundaries of mathematics, 50

computer science, economics and engineering. Analytical foundations for the techniques to solve the 51

constrained optimization problems involving continuous, differentiable functions and equality constraints 52

were already laid in the 18th century [6]. However, with advances in computing technology, constrained 53

optimization methods designed to handle a broader range of problems trace their origin to the 54

development of the simplex algorithm--the most commonly used algorithm to solve linear constrained 55

optimization problems--in 1947 [7-11]. Since that time, a variety of constrained optimization methods 56

have been developed in the field of operations research and applied across a wide range of industries. This 57

creates significant opportunities for the optimization of health care delivery systems and for providing 58

value by transferring knowledge from fields outside the health care sector. 59

In addition to capacity management, facility location, and efficient delivery of supplies, patient scheduling, 60

provider resource scheduling, and logistics are other substantial areas of research in the application of 61

constrained optimization methods to healthcare [12-16]. Constrained optimization methods may also be 62

very useful in guiding clinical decision-making in actual clinical practice where physicians and 63

patients face constraints such as proximity to treatment centers, health insurance benefit designs, 64

and the limited availability of health resources. 65

Constrained optimization methods can also be used by health care systems to identify the optimal 66

allocation of resources across interventions subject to various types of constraints [17-23]. These methods 67

have also been applied to disease diagnosis [24, 25], the development of optimal treatment algorithms 68

[26, 27], and the optimal design of clinical trials [28]. Health technology assessment using tools from 69

constrained optimization methods is also gaining popularity in health economics and outcomes research 70

[29]. 71

Recently, the ISPOR Emerging Good Practices Task Force on Dynamic Simulation Modeling Applications 72

in Health Care Delivery Research published two reports in Value in Health [30, 31] and one in 73

Pharmacoeconomics [32] on the application of dynamic simulation modeling (DSM) to evaluate problems 74

in health care systems. While simulation can provide a mechanism to evaluate various scenarios, by 75

design, they do not provide optimal solutions. The overall objective of the ISPOR Emerging Good 76

Practices Task Force on Constrained Optimization Methods is to develop guidance for health services 77

researchers, knowledge users and decision makers in the use of operations research methods to optimize 78

healthcare delivery and value in the presence of constraints. Specifically, this task force will (1) introduce 79

constrained optimization methods for conducting research on health care systems and individual-level 80

outcomes (both clinical and economic); (2) describe problems for which constrained optimization 81


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methods are appropriate; and (3) identify good practices for designing, populating, analyzing, testing and 82

reporting results from constrained optimization models. 83

The ISPOR Emerging Good Practices Task Force on Constrained Optimization Methods will produce two 84

reports. In this first report, we introduce readers to constrained optimization methods. We present 85

definitions of important concepts and terminology, and provide examples of health care decisions where 86

constrained optimization methods are already being applied. We also describe the relationship of 87

constrained optimization methods to health economic modeling and simulation methods. The second 88

report will present a series of case studies illustrating the application of these methods including model 89

building, validation, and use. 90

2. Definition of Constrained Optimization 91

92

Constrained optimization is a set of methods designed to efficiently and systematically find the best 93

solution to a problem characterized by a number of potential solutions in the presence of identified 94

constraints. It entails maximizing or minimizing an objective function that represents a quantifiable 95

measure of interest to the decision maker, subject to constraints that restrict the decision maker’s freedom 96

of action. Maximizing/minimizing the objective function is carried out by systematically selecting input 97

values for the decision from an allowed set and computing the objective function, in an iterative manner, 98

until the decision yields the best value for the objective function, a.k.a optimum. The decision that gives 99

the optimum is called the “optimal solution”. In some optimization problems, two or more different 100

decisions may yield the same optimum. Note that, programming and optimization are often used as 101

interchangeable terms in the literature, e.g., linear programming and linear optimization. Historically, 102

programming referred to the mathematical description of a plan/schedule, and optimization referred to 103

the process used to achieve the optimal solution described in the program. 104

105

The components of a constrained optimization problem are its objective function(s), its decision 106

variable(s) and its constraint(s). The objective function is a function of the decision variables that 107

represents the quantitative measure that the decision maker aims to minimize/maximize. Decision 108

variables are mathematical representation of the constituents of the system for which decisions are being 109

taken to improve the value of the objective function. The constraints are the restrictions on decision 110

variables, often pertaining to resources. These restrictions are defined by equalities/inequalities involving 111

functions of decision variables. They determine the allowable/feasible values for the decision variables. In 112

addition, parameters are constant values used in objective function and constraints, like the multipliers 113

for the decision variables or bounds in constraints. Each parameter represents an aspect of the decision-114

making context: for example, a multiplier may refer to the cost of a treatment. 115

3. A Simple Illustration of a Constrained Optimization Problem 116

117

Imagine you are the manager of a health care center, and your aim is to benefit as many patients as 118

possible. Let us say, for the sake of simplicity, you have two types of patients-- regular and severe patients, 119

and the demand for the health service is unlimited for both of these types. Regular patients can achieve 120

two units of health benefits and severe ones can achieve three units. Each patient, irrespective of severity, 121

takes 15 minutes for consultation; only one patient can be seen at any given point in time. You have one 122

hour of total time at your disposal. Regular patients require $25 of medications, and severe patients 123

require $50 of medications. You have a total budget of $150. What is the greatest health benefit this 124

center can achieve given these inputs and constraints? 125

At the outset, this problem seems straightforward. One might decide on four regular patients to use up all 126

the time that is available. This will achieve eight units of health benefit while leaving $50 as excess budget. 127

An alternate approach might be to see as many severe patients as possible since treating each severe 128

patient generates more per capita health benefits. Three patients (totaling $150) would generate 9 health 129


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units leaving 15 minutes extra time unused. There are other combinations of regular and severe patients 130

that would generate different levels of health benefits and use resources differently. 131

This is graphically represented in Figure 1, with regular patients on the x-axis and the severe patients on 132

the y-axis. Line CF is the time constraint limiting total time to one hour. Line BG is the budget constraint 133

limiting to $150. Any point to the south-west of these constraints (lines) respectively, will ensure that time 134

and budget do not exceed the respective limits. The combination of these together with non-negativity of 135

the decision variables, gives the feasible region. 136

The lines AB-BD-DF-FA form the boundary of the feasibility space, shown shaded in the figure. In 137

problems that are three or more dimensional, these lines would be hyperplanes. To obtain the optimal 138

solution, the dashed line is established, the slope depends on the relative health units of the two decision 139

variables (i.e., the number of regular and severe patients seen). This dashed line moves from the origin in 140

the north-east direction as shown by the arrow. The optimal solution is two regular patients and two 141

severe patients. This approach uses the entire one-hour time as well as the $150 budget. Since regular and 142

severe patients achieve two- and three-unit health benefits, respectively, we are able to achieve 10 units of 143

health benefit and still meet the time and budget constraints. 144

No other combination of patients is capable of achieving more benefits while still meeting the time and 145

budget constraints. Note that not all resource constraints have to be completely used to attain the optimal 146

solution. This hypothetical example is a small-scale problem with only two decision variables; the number 147

of regular and severe patients seen. Hence, they can be represented graphically with one variable on each 148

axis. 149

With the difficulty in representing larger problems graphically, we turn to mathematical approaches, such 150

as the simplex algorithm to find the solutions. The simplex algorithm is a structured approach of 151

navigating the boundary (represented as lines in two dimensions and hyperplanes in three or more 152

dimensions) of the feasibility space to arrive at the optimal solution. Table 1 summarizes the main 153

components of the example and notes several other dimensions of complexity (linear vs nonlinear, 154

deterministic vs stochastic, static vs dynamic, discrete/integer vs continuous) that can be incorporated 155

into constrained optimization models. 156

157


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Figure 1. Graphical Representation of Solving a Simple Integer Programming Problem 158

159

The mathematical formulation of the model is as follows: 160

161

Max fR xR + fL xL (objective function) 162

subject to cR xR + cL xL ≤ B (budget constraint) 163

tR xR + tL xL ≤ T (time constraint) 164

xR ,xL ≥ 0 and integer (decision variables) 165

166

Where: 167

cR,cL= cost of regular and severe patients, respectively 168

B = total budget available 169

tR,tL= time to see regular and severe patients, respectively 170

T = total time available 171

fR,fL= health benefits of regular and severe patients, respectively 172

xR,xL= number of regular and severe patients, respectively 173

174

In the current version of the problem, the parameters are: 175

fR = 2 health benefit units, fL = 3 health benefit units 176

cR = $25, cL = $50, B = $150 177

tR =0.25 hours, tL = 0.25 hours, T = 1 hour 178

179

So the model is as follows: 180

181

Max 2 xR + 3xL (objective function) 182

subject to 25xR + 50xL ≤ 150 (budget constraint) 183

0.25xR + 0.25xL ≤ 1 (time constraint) 184

xR ,xL ≥ 0 and integer 185

186

As described above, Figure 1 illustrates the graphical solution to this model. However, problems with 187

higher dimensionality must use mathematical algorithms to identify the optimal solution. The problem 188

described above falls into the category of linear optimization, because although the constraints and the 189

objective function are linear from an algebraic standpoint, the decision variables must be in the form of 190

integers. As it will be discussed further in section 5, there are other optimization modelling frameworks, 191

such as combinatorial, nonlinear, stochastic and dynamic optimization. 192

As the algorithms for integer optimization problems can take much longer to solve computationally than 193

those for linear optimization problems, one alternative is to set the integer optimization problem up and 194

solve it as a linear one. If fractional values are obtained, the nearest feasible integers can be used as the 195

final solution. This should be done with caution, however. First, rounding the solution to the nearest 196

integers can result in an infeasible solution or, and second, even if the rounded solution is feasible, it may 197

not be the optimal solution to the original integer optimization problem. Nonlinear optimization is 198

suitable when the constraints or the objective function are non-linear. In problems, where there is 199

uncertainty, such as the estimated health benefit of each patient might receive in the above example, 200

stochastic optimization techniques can be used. 201

Dynamic optimization (known commonly as dynamic programming) formulation might be useful when 202

the optimization problem is not static, that the problem context and parameters change in time and there 203

is an interdependency among the decisions at different time periods (for instance, when decisions made at 204

a given time interval, say number of patients to be seen now, affects the decisions for other time periods, 205


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such as the number of patients to be seen tomorrow). Table 1 summarizes the model components in the 206

hypothetical problem, relates it to health services with examples and identifies the specific terminology. 207

Table 1. Model Summary and Extensions 208

209

4. Problems That Can Be Tackled with Constrained Optimization Approaches 210

211

In this section, we list several areas within health care where constrained optimization methods have been 212

used in health services. The selected examples do not represent a comprehensive picture of this field, but 213

provide the reader a sense of what is possible. In Table 2, we compare problems using the terminology of 214

the previous section, with respect to decision makers, decisions, objectives, and constraints. 215

Table 2. Examples of Health Care Decisions for which Constrained Optimization is 216

Applicable 217

5. Steps in a Constrained Optimization Process 218

219

An overview of the main steps involved in a constrained optimization process [33] is described here and 220

presented in Table 3. Some of the steps are common to other types of modeling methods. It is important to 221

emphasize that the process of optimization is iterative, rather than comprising a strictly sequential set of 222

steps. 223

a) Problem structuring 224

225

This involves specifying the objective, i.e. goal, and identifying the decision variables, parameters and the 226

constraints involved. These can be specified using words, ideally in non-technical language so that the 227

optimization problem is easily understood. This step needs to be performed in collaboration with all the 228

relevant stakeholders, including decision makers, to ensure all aspects of the optimization problem are 229

captured. As with any modeling technique, it is also crucial to surface key modeling assumptions and 230

appraise them for plausibility and materiality. 231

b) Mathematical formulation 232

233

After the optimization problem is specified in words, it needs to be converted into mathematical notation. 234

The standard mathematical notation for any optimization problem involves specifying the objective 235

function and constraint(s) using decision variables and parameters. This also involves specifying whether 236

the goal is to maximize or minimize the objective function. The standard notation for any optimization 237

problem, assuming the goal is to maximize the objective, is as shown below: 238

Maximize z=f(x1, x2, …. xn, p1, p2, …. pk) 239

subject to 240

cj(x1, x2, …. xn, p1, p2, …. pk)≤Cj 241

for j=1,2,..m 242

where, x1, x2, …. xn are the decision variables, f(x1, x2, …. xn) is the objective function; and cj(x1, x2, …. xn, p1, 243

p2, …. pk)≤Cj represent the constraints. Note that the constraints can include both inequality and equality 244

constraints and that the objective function and the constraints also include parameters p1, p2, …. pk, which 245

are not varied in the optimization problem. Specification of the optimization problem in this mathematical 246

notation allows clear identification of the type (and number) of decision variables, parameters and the 247

constraints. Describing the model in mathematical form will be useful to support model development. 248

c) Model development 249

250


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The next step after mathematical formulation is model development. Model development involves solving 251

the mathematical problem described in the previous step, and often performed iteratively. The model 252

should estimate the objective function and the left hand side (LHS) values of the constraints, using the 253

decision variables and parameters as inputs. The complexity of the model can vary widely. Similar to 254

other types of modeling, the complexity of the model will depend on the outputs required, the level of 255

detail included in the model, whether it is linear or non-linear, stochastic or deterministic, static or 256

dynamic. 257

d) Perform model validation 258 259

As with any modeling, it is important to ensure that the model developed represents reality with an 260

acceptable degree of fidelity [33]. The requirements of model validation for optimization are more 261

stringent than for, for example, simulation models, due to the need for the model to be valid for all 262

possible combinations of the decision variables. Thus, appropriate caution needs to be taken to ensure 263

that the model assumptions are valid and that the model produces sensible results for the different 264

scenarios. At the very least, the validation should involve checking of the face validity (i.e. experts evaluate 265

model structure, data sources, assumptions, and results), and verification or internal validity (i.e. checking 266

accuracy of coding). 267

e) Select optimization method 268

This step involves choosing the appropriate optimization method, which is dependent on the type of 269

optimization problem that is addressed. Optimization problems can be broadly classified, depending 270

upon the nature of the objective functions and the constraints-for example, into linear vs non-linear, 271

deterministic vs stochastic, continuous vs discrete, or single vs multi-objective optimization. For instance, 272

if the objective function and constraints consist of linear functions only, the corresponding problem is a 273

linear optimization problem. Similarly, in deterministic optimization, the parameters used in the 274

optimization problem are fixed while in stochastic optimization, uncertainty is incorporated. Optimization 275

problems can be continuous (i.e. decision variables are allowed to have fractional values) or discrete (for 276

example a hospital ward may be either open or closed; the number of CT scanners which a hospital buys 277

must be a whole number). 278

Most optimization problems have a single objective function, however when optimization problems have 279

multiple conflicting objective functions, they are referred to as multi-objective optimization problems. The 280

optimization method chosen needs to be in line with the type of optimization problem under 281

consideration. Once the optimization problem type is clear (e.g. discrete or nonlinear), a number of texts 282

may be consulted for details on solution methods appropriate for that problem type [33-36]. 283

Broadly speaking, optimization methods can be categorized into exact approaches and heuristic 284

approaches. Exact approaches iteratively converge to an optimal solution. Examples of these include 285

simplex methods for linear programming and the Newton method or interior point method for non-linear 286

programming [34, 37]. Heuristic approaches provide approximate solutions to optimization problems 287

when an exact approach is unavailable or is computationally expensive. Examples of these techniques 288

include relaxation approaches, evolutionary algorithms (such as genetic algorithms), simulated annealing, 289

swarm optimization, ant colony optimization, and tabu-search. Besides these two approaches (i.e. exact or 290

heuristic), other methods are also available to tackle large-scale problems as well (e.g. decomposition of 291

the large problems to smaller sub-problems). 292

There are software programs that help with optimization; interested readers are referred to the website of 293

INFORMS (www.informs.org) for a list of optimization software. The users need to specify, and more 294

importantly understand, the parameters used as an input for these optimization algorithms (e.g., the 295

termination criteria such as the level of convergence required or the number of iterations). 296

http://www.informs.org/


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f) Perform optimization/sensitivity analysis 297

Optimization involves systematically searching the feasible region for values of decision variables and 298

evaluating the objective function, consecutively, to find a combination of decision variables that achieve 299

the maximum or minimum value of the objective function, using specific algorithms. Once the 300

optimization algorithm has finished running, in some cases, the identified solution can be checked to 301

verify that it satisfies the “optimality conditions” (i.e. Karush-Kuhn-Tucker conditions) [38], which are the 302

mathematical conditions that define the optimality. Once the optimality is confirmed, the results need to 303

be interpreted. 304

First, the results should be checked to see if there is actually a feasible solution to the optimization 305

problem, i.e. whether there is a solution that satisfies all the constraints. If not, then the optimization 306

problem needs to be adjusted, (e.g., relaxing some constraints or adding other decision variables) in order 307

to broaden the feasible solution space. If a feasible optimal solution has been found, the results need to be 308

understood – this involves interpretation of the results to check whether the optimal solution, i.e., values 309

of decision variables, constraints and objective function makes sense. 310

It is also good practice to repeat the optimization with different sets of starting decision variables to 311

ensure the optimal solution is the global optimum rather than local optimum. Sometimes, there may be 312

multiple optimal solutions for the same problem (i.e. multiple combinations of decision variables that 313

provide the same optimal value of objective function). For multi-objective optimization problems (i.e. 314

problems with two or more conflicting objectives), Pareto optimal solutions are constructed from which 315

optimal solution can be identified based on the subjective preferences of the decision maker [39, 40]. 316

It is good practice to run the optimization problem using different values of parameters, in order to verify 317

the robustness of the optimization results. Sensitivity analysis is an important part of building confidence 318

in an optimization model, addressing the structural and parametric uncertainties in the model by 319

analyzing how the decision variables and optimum value react to changes in the parameters in the 320

constraints and objective function, which ensures that the optimization model and its solution are good 321

representations of the problem at hand. 322

Sometimes a solution may be the mathematically optimal solution to the specified mathematical problem, 323

but may not be practically implementable. For example, the “optimal” set of nurse rosters may be 324

unacceptable to staff as it involves breaking up existing teams, deploying staff with family responsibilities 325

on night shifts, or reducing overtime pay to level where the employment is no longer attractive. Analysts 326

should resist the temptation to spring their optimal solution on unsuspecting stakeholders, expecting 327

grateful acceptance: rather, those affected by the model should be kept in the loop through the modeling 328

process. The optimal solution may come as a surprise: it is important to allow space in the modeling 329

process to explore fully and openly concerns about whether the “optimal” solution is indeed the one the 330

organization should implement. 331

g) Report results 332

333

The final optimal solution, and if applicable, the results of the sensitivity analyses should be reported. This 334

will include the results of the optimum ‘objective function’ achieved and the set of ‘decision variables’ at 335

which the optimal solution is found. Both the numerical values (i.e. the mathematical solution) and the 336

physical interpretation, i.e., the non-technical text describing the meaning of numerical values, should be 337

presented. The optimal solution identified can be contextualized in terms of how much ‘better’ it is 338

compared to the current state. For example, the results can be presented as improvement in benefits such 339

as QALYs or reduction in costs. 340

341


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It is often necessary to report the optimization method used and the results of the ‘performance’ of the 342

optimization algorithm, e.g., number of iterations to the solution, computational time, convergence level, 343

etc. This is important as it helps users understand whether a particular algorithm can be used “online” in a 344

responsive fashion, or only when there is significant time available, e.g. in a planning context. Dashboards 345

can be useful to visualize these benefits and communicate the insights gained from the optimal solution 346

and sensitivity analyses. 347

348

h) Decision making 349

350

The final optimal solution and its implications for policy/service reconfiguration should be presented to all 351

the relevant stakeholders. This typically involves a plan for amending the ‘decision variables’, (e.g., shift 352

patterns, screening frequency--see Table 2 for examples of decision variables--to those identified in the 353

optimal solution). Before an optimal solution can be implemented, it will require getting the ‘buy-in’ from 354

the decision makers and all the stakeholders, e.g., frontline staff such as nurses, hospital managers, etc., to 355

ensure that the numerical ‘optimal’ solution found can be operationalized in a ‘real’ clinical setting. It is 356

important to have the involvement of decision makers throughout the whole optimization process to 357

ensure that it does not become a purely numerical exercise, but rather something that is implemented in 358

real life. After the decision is made, data should still be collected to assess the efficiency and demonstrate 359

the benefits of the implementation of the optimal solution. 360

361

If decision makers are not directly involved in model development they may choose not to implement the 362

“optimal” solution as it comes from the model. This is because the model may fail to capture key aspects 363

of the problem (for example, the model may maximize aggregate health benefits but the decision maker 364

may have a specific concern for health benefits for some disadvantaged subgroup). This does not 365

(necessarily) mean that the optimization modeling has not been useful – enabling a decision maker to see 366

how much health benefit must be sacrificed to satisfy her equity objective may prove to be beneficial 367

towards the overall objective. After the decision is made the story does not come to an end: data should 368

continue to be collected to demonstrate the benefits of whatever solution is implemented, as well as 369

guiding future decision making. 370

371

Table 3 presents the two different stages in optimization i.e. the modeling stage and optimization stage, 372

highlighting that model development is necessary before optimization can be performed. The goal of 373

constrained optimization is to identify an optimal solution that maximizes or minimizes a particular 374

objective subject to existing constraints. 375

Table 3. Steps in an Optimization Process 376

377

6. Relationship of Constrained Optimization to Related Fields 378

379

The use of constrained optimization can be classified into two categories. The first category is the use of 380

constrained optimization as a decision-making tool. The simple illustration in section 3 and all the 381

examples in section 4 are considered to fall under this category. The second category is the use of 382

constrained optimization as an auxiliary analysis tool. In this category, optimization is an embedded tool 383

and the results of which are often not the end results of a decision problem, but rather they are used as 384

inputs for other analysis/modeling methods (e.g. optimization used in the multiple criteria decision 385

making; in calibrating the inputs for health economic or dynamic simulation models; in machine learning 386

and other statistical analysis methods like solving regression models or propensity score matching). 387

As a decision-making tool, optimization is complementary to other modeling methods such as health 388

economic modeling, simulation modeling and descriptive, predictive (e.g. machine learning) and 389


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prescriptive analytics. Most modeling methods typically only evaluate a few different scenarios and 390

determine a good scenario within the available options. In contrast, the aim of optimization methods is to 391

efficiently identify the best solution overall, given the constraints. In the absence of using optimization 392

methods, a brute force approach, in which all possible options are sequentially evaluated and the best 393

solution is identified among them, might be possible for some problems. However, for most problems, it is 394

too complex and too time consuming to identify and evaluate all possible options. Optimization methods 395

and heuristic approaches might use efficient algorithms to identify the optimal solution quickly, which 396

would otherwise be very difficult and time consuming. 397

Also, model development using these other methods might be necessary before optimization, especially in 398

situations where the objective function or constraints cannot be represented in a simple functional form. 399

Thus, all models currently used in health care such as health economic models, dynamic simulation 400

models and predictive analytics (including machine learning) can be used in conjunction with 401

optimization methods. 402

a) Constrained Optimization Methods Compared with Traditional Health Economic 403

Modeling in Health Technology Assessments 404

405

Constrained optimization methods differ substantially from health economic modeling methods 406

traditionally used in health technology assessment processes [41]. The main difference between the two 407

approaches is that traditional health economic modeling approaches, such as Markov models, are built to 408

estimate the costs and effects of different diagnostic and treatment options. If decision makers are basing 409

their judgements on modeling results, they may not formally consider the constraints and resource 410

implications in the system. Constrained optimization methods provide a structured approach to optimize 411

the decision problem and to present the best alternatives given an optimization criterion, such as 412

constrained budget or availability of resources. 413

These differences have major implications. There is an opportunity to learn from optimization methods to 414

improve Health Technology Assessment (HTA) processes [42-46]. Optimization is a valuable means of 415

capturing the dynamics and complexity of the health system to inform decision making for several 416

reasons. Constrained optimization methods can: 417

i. Explicitly take budget constraints into account - Informed decision making about resource 418

allocation requires an external estimate of the decision-maker’s willingness to pay for a unit of 419

health outcome – the threshold. Decision making based on traditional health economic models 420

then relies on the principle that by repeatedly applying the threshold to individual HTA decisions, 421

optimization of the allocation of health resources will be achieved. 422

423

However, the focus of health economics (HE) is usually about relative efficiency without explicit 424

consideration of budget because many jurisdictions do not explicitly implement a constrained 425

budget nor do they employ mechanisms to evaluate retrospectively cost-effectiveness of medical 426

technologies currently in use. 427

ii. Address multiple resource constraints in the health system, such as resource capacity: Constrained 428

optimization methods also allow consideration of the effect of other constraints in the health 429

system, such as capacity or short-term inefficiencies. Capacity constraints are usually neglected in 430

health economic models. In HE models, the outcomes are central to decision makers while the 431

process to arrive at these outcomes is most of the time ignored. 432

433

For health policy makers and health care planners, such capacity considerations are critical and 434

cannot be neglected. Likewise, some technologies are known for short-term inefficiencies, e.g., 435

large equipment such as PET-MR imaging, are usually not taken into consideration. It takes a 436


12

certain amount of time before a new device operates efficiently, and such short-term inefficiencies 437

do influence implementation [47]. 438

iii. Account for system behavior and decisions over time: Traditional health economic models are 439

often limited to informing a decision of a single technology at a single point in time. Health 440

economic models with a clinical perspective, such as a whole disease model [48, 49], or a 441

treatment sequencing model, may allow the full clinical pathway to be framed as a constrained 442

optimization problem that accounts for both intended and unintended consequences of health 443

system interventions over time with feedback mechanisms in the system. 444

445

Each combination of decisions within the pathway can be a potential solution, constrained by the 446

feasibility of each decision, e.g., the licensed indication for various treatments within a clinical 447

pathway. These whole disease and treatment sequencing models can evaluate alternative guidance 448

configurations and report the performance in terms of an objective function (cost per QALY, net 449

monetary benefit) [50, 51]. 450

iv. Inform decision makers about implementability of solutions that are recommended: Health 451

economic models are not typically constrained – it is assumed that resources are available as 452

required and are thus affordable, similarly the evidence used in the models come from controlled 453

clinical settings, which are idealized settings compared to real clinical setting. An advantage of 454

constrained optimization is the ability to obtain optimal solutions to decision problems and have 455

sensitivity analyses performed. Such analyses inform decision makers about alternate realistic 456

solutions that are feasible and close to the optimal solution. 457

Thus, in some sense, classic health economics models are ‘hypothetical’ to illustrate the potential value as 458

measured by a specific outcome with respect to cost, whereas optimization is focused on what can be 459

achieved in an operational context. This suggests constrained optimization methods have great value for 460

informing decisions about the ability to implement a clinical intervention, program, or policy as they 461

actually consider these constraints in the modeling approach. 462

463

b) Constrained Optimization Methods Compared with Dynamic Simulation Models 464 465

Dynamic simulation modeling methods (DSMs), such as system dynamics, discrete event simulation and 466

agent based modeling are used to design and develop mathematical representations, i.e., formal models, of 467

the operation of processes and systems. They are used to experiment with and test interventions and 468

scenarios and their consequences over time in order to advance the understanding of the system or 469

process, communicate findings, and inform management and policy design [30-32, 52-54]. These 470

methods have been broadly used in health applications [55-57]. 471

Unlike constrained optimization methods, DSMs do not produce a specific solution. Rather they allow for 472

the evaluation of a range of possible or feasible scenarios or intervention options that may or may not 473

improve the system’s performance. Constrained optimization methods, in general, seek to provide the 474

answer to which of those options is the “best”. Hence, the types of problems and questions that can be 475

addressed with DSMs [30-32] are different from those that are addressed with optimization methods. 476

However, both types of methods can be complementary to each other in helping us to better understand 477

systems. 478

Traditionally, constrained optimization methods have served two distinct purposes in DSM development. 479

1) model calibration – fitting suitable model variables to past time series is discussed elsewhere [30-32]; 480

2) evaluating a policy’s performance/effect relative to a criterion or set of criteria. However, the 481

complexity of DSMs compared to simple analytic models may render exact constrained optimization 482

approaches cumbersome, inappropriate and potentially infeasible due to the large search space e.g., using 483

methods of optimal control. 484


13

Due to this complexity, alternatives to exact approaches such as heuristic search strategies are available. 485

Historically, these types of methods have been used in system dynamics and other DSMs. Due to their 486

heuristic nature, there is no certainty of finding the “best” or optimal parameter set rather “good enough” 487

solutions. Hence, the ranges assigned need careful consideration in order to get “good” solutions, i.e., 488

prior knowledge of sensible ranges both from knowledge about the system and knowledge gained from 489

model building. 490

Optimization is used as part of system dynamics to gain insight about policy design and strategy design, 491

particularly when the traditional analysis of feedback mechanisms becomes risky due to the large numbers 492

of loops in a model [58]. Similar procedures to evaluate policies and strategies can be can be utilized in 493

discrete event simulation (DES) and agent based modeling (ABM), e.g., simulated annealing algorithms 494

and genetic algorithms. 495

c) Constrained Optimization Methods as Part of Analytics 496

Constrained optimization methods fall within the area of analytics as defined by the Institute for 497

Operations Research and the Management Sciences (INFORMS, https://www.informs.org/Sites/Getting-498

Started-With-Analytics). Analytics can be classified into: descriptive, predictive and prescriptive analytics 499

(Figure 2), and discussed below. Constrained optimization methods are a special form of prescriptive 500

analytics. 501

i. Descriptive analytics concern the use of historical data to describe a phenomenon of interest—502 with a particular focus on visual displays of patterns in the data. Descriptive analytics is 503 differentiated from descriptive analysis which uses statistical methods to test hypotheses about 504 relationships among variables in the data. Health services research typically uses theory and 505 concepts to identify hypotheses, and historical data are used to test these hypotheses using 506 statistical methods. Examples may include natural history of aging, disease progression, 507 evaluation of clinical interventions, policy interventions, and many others. Traditional health 508 services for the most part falls within the area of descriptive analytics. 509

ii. Predictive analytics and machine learning focus on forecasting the future states of disease or 510 states of systems. With the increased volume and dimensions of health care data, especially 511 medical claims and electronic medical record data, and the ability to link to other information 512 such as feeds from personal devices and socio demographic data, big data methods such as 513 machine learning are garnering increased attention [59]. 514 Machine learning methods, such as predictive modeling and clustering, have an important 515

intersection with constrained optimization methods. Machine learning methods are valuable 516

for addressing problems involving classification, as well as data dimension reduction issues. 517

And maybe most importantly, optimization often needs forecasts and estimates as inputs, 518

which can be obtained from the results of machine learning algorithms. A discussion of 519

machine learning methods is beyond the scope of this paper. 520

However, the interested reader will find a detailed introduction elsewhere [60, 61]. Machine 521

learning has the ability to “mine” data sets and discover trends or patterns. These are often 522

valuable to establish thresholds or parameter values in optimization models, where it is 523

otherwise difficult to determine the values. Constrained optimization can also leverage the 524

ability of machine learning to reduce high dimensionality of data, say with thousands or 525

millions of variables to key variables. 526

iii. Prescriptive analytics uses the understanding of systems, both the historical and future based 527 on historical (descriptive) and predictive analytics respectively to determine future course of 528 action/decisions. Traditional (without optimization) clinical trials and interventions fall under 529 the category of prescriptive analytics (“Change what will happen” in figure). Constrained 530 optimization is a specialized form of prescriptive analytics, since it helps with determining the 531 optimal decision or course of action in the presence of constraints 532 (https://www.informs.org/Sites/Getting-Started-With-Analytics/Analytics-Success-Stories). 533

534

https://www.informs.org/Sites/Getting-Started-With-Analytics

https://www.informs.org/Sites/Getting-Started-With-Analytics


14

Figure 2. Descriptive, Predictive, and Prescriptive Analytics. 535

536

7. Summary and Conclusions 537

538

This is the first report of the ISPOR Constrained Optimization Methods Emerging Good Practices Task 539

Force. It introduces readers to the application of constrained optimization methods to health care systems 540

and patient outcomes research problems. Such methods provide a means of identifying the best policy 541

choice or clinical intervention given a specific goal and given a specified set of constraints. Constrained 542

optimization methods are already widely used in health care in areas such as choosing the optimal location 543

for new facilities, making the most efficient use of operating room capacity, etc. 544

However, they have been less widely used for decision making about clinical interventions for patients. 545

Constrained optimization methods are highly complementary to traditional health economic modeling 546

methods and dynamic simulation modeling—providing a systematic and efficient method for selecting the 547

best policy or clinical alternative in the face of large numbers of decision variables, constraints, and 548

potential solutions. As health care data continues to rapidly evolve in terms of volume, velocity, and 549

complexity, we expect that machine learning techniques will also be increasingly used for the development 550

of models that can subsequently be optimized. 551

In this report, we introduce readers to the vocabulary of constrained optimization models and outline a 552

broad set of models available to analysts for a range of health care problems. We illustrate the 553

formulation of a linear program to maximize the health benefit generated in treating a mix of “regular” 554

and “severe” patients subject to time and budget constraints and solve the problem graphically. Although 555

simple, this example illustrates many of the key features of constrained optimization problems that would 556

commonly be encountered in health care. 557

In the second task force report, we describe several case studies that illustrate the formulation, estimation, 558

evaluation, and use of constrained optimization models. The purpose is to illustrate actual applications of 559

constrained optimization problems in health care that are more complex than the simple example 560

described in the current paper and make recommendations on emerging good practices for the use of 561

optimization methods in health care research. 562

563


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Optimization TFR Acknowledgements 683

The co-authors thank all those who commented orally during task force workshop presentations at 684

ISPOR DC & Vienna meetings. The co-authors gratefully acknowledge the 22 reviewers who 685

contributed their time and expertise through submission of written comments: Ekkehard Beck, Bjorn 686

Berg, J. Jaime Caro, Koen Degeling, Chris Hane, Julie L. Higle, Ng Chin Hui, Dawn Lee, Nan Liu, 687

Lena Burgos Liz, Nikolas Popper, Farhan Abdul Rauf, Emily S. Reese, Ajit Singh, Eunju Todd, 688

Pepijn Vemer, Amir Viyanchi, Richard J. Willke, Beth Woods, Greg Zaric. Their generous feedback 689

improved the manuscript and made it an expert consensus ISPOR Task Force Report. 690

Special thanks to Bhash Naidoo, Senior Technical Adviser (Health Economics) – NICE Centre for 691

Guidelines, National Institute for Health and Care Excellence, London, UK for his comments. 692

693

Finally, we are especially grateful to ISPOR staff, including Elizabeth Molsen, for helping us get these 694

task force reports done - from beginning to end - and to Kelly Lenahan for her assistance in producing 695

this report. 696

697

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Figure 1. Graphical Representation of Solving a Simple Integer Programming Problem 699


18

700

701

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Figure 2. Descriptive, Predictive, and Prescriptive Analytics. 703

704

705

706

707

Wilson, ISPOR 2014

Future states

Optimal decision

Optimization


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708

Table 1. Model Summary and Extensions 709

710

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Hypothetical problem Real-life Health Services Terminology

Aim Maximize health/health

care benefits

Maximize health/health care

benefits

Objective

function

Options

available

Regular or severe patients Service lines, case mix, service

mix, etc.

Decision

variables

Constraints Total cost < $15

Total time < 1 hour

Budget constraint

Time constraint

Resource constraint (e.g. staff,

beds, etc.)

Constraints

Evidence

base

Cost of each patient,

health benefits of each

patient and the time taken

for consultation

Costs, health benefits, and

other relevant data associated

with each intervention to be

selected

Model (to

determine the

objective

function and

constraints)

Complexity Static

The problem does not

have a time component;

decision made in one time

period does not affect

decisions made in another

Deterministic

All the information is

assumed to be certain (e.g

Cost of each patients,

health benefits of each

patient and the time taken

for consultation)

Linear (i.e. each

additional patient costs

the same and achieves

same health benefits)

Integer/discrete

The decision variables

(number of patients) can

only take discrete and

integer values

Dynamic

The optimization problem and

parameters may change in

different time points, and the

decision made at any point in

time can affect decisions at

later time points (e.g. there can

be a capacity constraint defined

on 2 months, whereas the

planning cycle is 1 month)

Stochastic

Know that the information is

uncertain (i.e. uncertainty in

the costs and benefits of the

interventions)

Non-linear (objective function or constraints may have a non-linear relationship with the model parameters, e.g. total costs and QALYs typically have a non-linear relationship with the model parameters) Continuous

The decision variables can take

fractional values (e.g. number

of hours)

Optimization

method


20

738

739

Table 2. Examples of Health Care Decisions for which Constrained Optimization is 740

Applicable 741

Type of health care problem

Typical decision makers

Typical decisions Typical objectives

Typical constraints

Resource allocation within and across disease programs

Health authorities, insurance funds

List of interventions to be funded

Maximize population health

Overall health budget, other legal constraints for equity

Resource allocation for infectious disease management

Public health agencies, health protection agencies

Optimal vaccination coverage level

Minimize disease outbreaks and total costs

Availability of medicines, disease dynamics of the epidemic

Allocation of donated organs

Organ banks, transplant service centers

Matching of organs and recipients

Maximize matching of organ donors with potential recipients

Every organ can be received by at most one person

Radiation treatment planning

Radiation therapy providers

Positioning and intensity of radiation beams

Minimizing the radiation on healthy anatomy

Tumor coverage and restriction on total average dosage

Disease management models

Leads for a given disease management plan

Best interventions to be funded, best timing for the initiation of a medication, best screening policies

Identify the best plan using a whole disease model, maximizing QALYs

Budget for a given disease or capacity constraints for healthcare providers

Workforce planning/ Staffing / Shift template optimization

Hospital managers, all medical departments (e.g., ED, nursing)

Number of staff at different hours of the day, shift times

Increase efficiency and maximize utilization of healthcare staff

Availability of staff, human factors, state laws (e.g., nurse-to-patient ratios), budget

Inpatient scheduling

Operation room/ ICU planners

Detailed schedules Minimize waiting time

Availability of beds, staff

Outpatient scheduling

Clinical department managers

Detailed schedules Minimize over- and under-utilization of health care staff

Availability of appointment slots

Hospital facility location

Strategic health planners

Set of physical sites for hospitals

Ensure equitable access to hospitals

Maximum acceptable travel time to reach a hospital

742

Table 3. Steps in an Optimization Process 743

Stage Step Description

Modeling

Problem structuring Specify the objective and constraints, identify decision variables and parameters, and list and appraise model assumptions

Mathematical formulation Present the objective function and constraints in mathematical notation using decision variables and


21

parameters Model development Develop the model; representing the objective function

and constraints in mathematical notation using decision variables and parameters

Model validation Ensure the model is appropriate for evaluating all possible scenarios (i.e. different combinations of decision variables and parameters)

Optimization

Select optimization method

Choose an appropriate optimization method and algorithm based on the characteristics of the problem

Perform optimization/sensitivity analysis

Use the optimization algorithm to search for the optimal solution and examine performance of optimal solution for reasonable values of parameters

Report results Report the results of optimal solution and sensitivity analyses

Decision making

Interpret the optimal solution and use it for decision making

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